Dyson’s Model in Infinite Dimensions Is Irreducible

Abstract

Dyson’s model in infinite dimensions is a system of Brownian particles interacting via a logarithmic potential with an inverse temperature of $$ \beta = 2$$ . The stochastic process is given as a solution to an infinite-dimensional stochastic differential equation. Additionally, a Dirichlet form with the sine $$ _2$$ point process as a reference measure constructs the stochastic process as a functional of the associated configuration-valued diffusion process. In this paper, we prove that Dyson’s model in infinite dimensions is irreducible.